Interaction of a supersonic particle with a three-dimensional complex plasma
E. Zaehringer,a) M. Schwabe, S. Zhdanov, D. P. Mohr, C. A. Knapek, P. Huber, I. L. Semenov, and H. M. Thomas
Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 82234 Weßling, Germany
(Dated: 24 February 2018)
The influence of a supersonic projectile on a three-dimensional complex plasma is studied. Micron sized particles in a low-temperature plasma formed a large undisturbed system in the new ”Zyflex” chamber during microgravity conditions. A supersonic probe particle excited a Mach cone with Mach number M ≈ 1.5 − 2 and double Mach cone structure in the large weakly damped particle cloud. The speed of sound is measured with different methods and particle charge estimations are compared to calculations from standard theories. The high image resolution enables the study of Mach cones in microgravity on the single particle level of a three-dimensional complex plasma and gives insight to the dynamics. A heating of the microparticles is discovered behind the supersonic projectile but not in the flanks of the Mach cone.
PACS numbers: 52.35.Dm
Keywords: Dusty Plasma, Complex Plasma, Supersonic motion
Mach cones are waves introduced by supersonic projec-tiles. It is well understood that supersonic flows across the surface of airplanes1and projectiles2,3generate heat. In fluids the Mach cones are shock waves, in complex plasmas they are compressional waves in the majority of cases. Here, we use the advantage of complex plas-mas that individual particles are traceable to study in detail the effect of a supersonic projectile on the sin-gle particle dynamics. Complex plasmas are plasmas
FIG. 1. (Color online) Mach cone behind a supersonic parti-cle. Overlay of 6 consecutive images at 83 fps of the supersonic particle and surrounding microparticles with a field of view (FoV) of 14.1 × 7.6 mm2. Each image has an exposure time
of 10 ms and a dead time of ≈ 2 ms. Each color represents two consecutive frames and shows trajectories of ≈ 22 ms. The colors are chronologically ordered from red over green to blue. White vertical lines indicate the location of Fig. 2 a).
a)Electronic mail: email@example.com
filled with well-defined, mostly monodisperse microparti-cles which then form a system that can possess gaseous, liquid or solid properties4–10. The microparticles inter-act by Yukawa potentials11 and the resulting strongly coupled system shows a lot of fascinating soft matter properties, such as melting12,13, vortices14, dust den-sity waves15, and self-propulsion16. Ground-based exper-iments mostly produce two dimensional (2D) systems. It is possible to observe three dimensional (3D) clouds of complex plasma in microgravity conditions on parabolic flights or on board the International Space Station (ISS). Sometimes, extra particles (probe particles or projec-tiles) move through the complex plasma and generate disturbances which offer additional possibilities to study the system17. Wakes, resp. Mach cones for supersonic velocities, formed behind these extra particles are ex-cited by randomly occurring, additional particles18–20, particle guns21,22 and lasers23–25, and are often used as diagnostics, see Ref. 20, 22, 26–29. Mach cones are also topics of numerous theoretical works30–35. Mach cones were found by chance in a 2D complex plasma system in 199918,19, then the research was refined by laser techniques23. This way compressional23, shear24, and combined25 Mach cones were studied to gain de-tailed knowledge of the underlaying effects, such as the velocity fields. In 2009 Ref. 28 reported the first com-pressional Mach cone under microgravity generated by a spontaneously accelerated, supersonic particle. Super-sonic projectiles with multiple cone structure are possi-ble due to the elastic properties of strongly coupled sys-tems. Neutral gas friction limits the number of Mach cones by damping the amplitude of the alternating com-pression and expansion zones19. In the study presented here, a fast projectile particle with supersonic speed was recorded in experiments performed in the framework of the Ekoplasma project36. The probe particle generated
60 90 120 60 90 120 60 90 120 I (a .u.) −100 −50 0 50 100 d (px) 60 90 120 a) b) c) d) e) f) c)d) e) f) θ
FIG. 2. (Color online) Example of the measurement with method 4. a) Overlay of 9 consecutive images at 83 fps of the supersonic particle with a FoV of 7.2 × 7.6 mm2. The image
was blurred to increase the contrast of the double Mach cone structure. The path of the supersonic particle is marked as thick white line, thin lines in white indicate the locations of the perpendicular profiles. b) Sketch of the Mach angle θ (yellow in the colored figure). c)-f) Pixel intensity I against the distance d in pixel (px) along the perpendicular profiles from a). Gray curves are full profile, black vertical lines show the positions of the automatically detected first maxima.
a fine structured double Mach cone in the 3D complex plasma system20 during a parabolic flight.
The Ekoplasma team participated in the 29th parabolic flight campaign of the German Aerospace Cen-ter (DLR) to attain microgravity conditions. The heart of the facility consists of the ”Zyflex” chamber, which is a capacitively coupled plasma chamber with ≈ 12 cm electrode diameter. Further details on the setup are de-scribed in Ref. 36 and 37. The electrodes can be moved and were set to a distance of 60 mm in the experiment. They excited the 13.56 MHz rf-discharge with 30.6±0.4 V voltage peak-to-peak between upper and lower elec-trodes. 4.41 ± 0.08 µm diameter melamine-formaldehyde (MF) microparticles were dispensed in the argon plasma under a low neutral gas pressure of 2.85 Pa during micro-gravity conditions. The microparticle cloud was illumi-nated by a 660 nm laser sheet with 0.16 − 0.25 mm width and observed by several cameras with a maximal reso-lution of 23.5 µm/px and a frame rate of 83 fps. During one parabola, a supersonic particle (the ”probe parti-cle” in the following) was observed to move through a relatively stable microparticle cloud. The interaction of the supersonic particle with the complex plasma cloud was observed for about 100 frames over a timespan of ≈ 1.2 s. The probe particle appeared in the upper part of the cloud, moved downwards and disappeared.Then it reappeared for 39 frames (≈ 0.48 s) moving downwards with supersonic speed. Afterwards, the particle left the complex plasma cloud. The video of the event is attached
TABLE I. Results from the Mach relation for the speed of sound cs and its statistic uncertainty ∆cs (other errors are neglected) for the different methods proposed: 1) Manual measurement 2) Cross-correlation with fixed rotation center 3) Cross-correlation with free center 4) Cone detection in per-pendicular profiles
Results from the methods 1 2 3 4
cs (mm/s) 20.4 21.0 18.7 20.6
∆cs (mm/s) 0.1 0.1 0.3 0.4
in the supplementary material. As in Ref. 23, two cone shaped waves are visible due to weak damping from the low neutral gas pressure and the strong interactions in the coupled system. The path was straight enough to observe the probe particle with its wave fronts during 39 consecutive images of the sequence. Thus, we can assume the movement to be parallel to the laser sheet and mainly downwards, see Fig. 1. The whole event was captured on the single particle level. For measurements of the Mach angle, the contrast was increased by blurring 9 consec-utive images with a radius of 5 px and overlaying them centered at the probe particle position, see Fig. 2 a).
The Mach relation describes the relation between the speed of sound cs and the probe particle speed vp with
the Mach angle θ, see Fig. 2 b):
cs/vp= sin θ. (1)
It is used to gain information about the properties of the microparticle system. Measuring vp and θ results in a
measurement of the speed of sound cs20. From the sound
speed and the plasma parameters we can infer the par-ticle charge by using different charging models, such as Ref. 38. The determination of the probe particle speed is easily done by particle tracking velocimetry (PTV)39, while the Mach angle is often measured manually in the literature19,26,29, which has the disadvantage of subjec-tive perception. We will demonstrate different methods to measure the Mach angle in the following. All methods were applied on a blurred overlay of 9 consecutive images as described above to get a better visibility of the Mach cone.
Method 1: Manual measurement We drew 10 lines along each flank of the cone and measured their angle with the image processing program ImageJ40. The dif-ference of the angles yields the double Mach angle. This leads to underestimated errors, mainly due to the ne-glect of systematic errors since there is only the per-ception of one analyst. The result is a sound speed
cs = 20.4 ± 0.1 mm/s. Estimating the systematic error
would be possible by repeating the measurement with different analysts, but this would increase the time and effort of this method. Computer based methods are a
more reproducible and more automatic way to do the measurement.
Method 2: Cross-correlation with fixed rotation center
In method 2, the Mach angle θ was measured by cross-correlation with a white rectangle, as in Ref. 20. The white rectangle is rotated around a rotation point in the image such as the center of the probe particle, to find two angles with maximal cross-correlation. Moving the rotation point further away on the probe particle tra-jectory decreases the measured angle systematically and vice versa. Due to this critical effect, the rotation point was chosen by finding the cross-section of the Mach cone flanks from the data of method 1 to be in the distance of ≈ 40 px in front of the probe particle. The rectan-gle width was changed from 10 to 30 px in steps of 5 px to get better statistics. The Mach relation then yields
cs= 21.0 ± 0.1 mm/s. The error was calculated from this
average and is small for consistent measurements. The systematic error from the choice of the rotation center is neglected (≈ 0.6◦per pixel offset along the probe particle trajectory) and therefore the error seems to be underesti-mated. Also, this method only uses the compression zone of the first Mach cone with high particle density and high image intensity, not the expansion zone with low parti-cle density and image intensity. In the last 10 frames of the sequence the probe particle leaves the microparticle cloud, therefore the compression zone disappears faster compared to the expansion zone. This leads to scattered results at higher velocities, see supp. Fig. b).
Method 3: Cross-correlation with free center The method from Ref. 37 was used as a third way to mea-sure the Mach angle. By using method 3 it is possible to avoid having to choose a rotation center as in method 2. Four different masks for each Mach cone flank from the first and the second cone were generated by broadly redrawing the flank in black-and-white. Then, the al-gorithm searches for the maximum cross-correlation of the black-and-white test-pattern on the binarized image, see Ref. 37 for details. The value is saved according to the particular mask if the test-pattern is located within it. The correlation curves for each flank are analyzed in a predefined angle range in the region where the values are higher than 80% of the maximal value. If infliction points exist close to the maximum, the range is restricted to them. Afterwards, the values are mapped onto the in-terval [0.1, 1]. A Gaussian distribution is fitted to these values and one obtains an angle (center) with uncertainty (standard deviation) for each flank. The difference of the angles of the first cone results in 2θ, giving the Mach re-lation. Method 3 is the most complex of the presented methods. The results depend on the correct choice of the binarization and of the test-pattern (width and length).
Method 4: Cone detection in perpendicular profiles
The last method uses the brightness profile perpendicu-lar to the probe particle trajectory in order to determine the points on the Mach cone flanks. The Mach angle is then calculated by the inclination of linear fits through the points of each flank, see Fig. 2. Our algorithm
first searches for the first maximum of the profile. A polynomial of order 3 is fitted onto the first data peak and the position of the maximum is calculated. See Fig. 2 for exemplary profiles and resulting positions of the maximums. Each profile results in two points in the image, one for each Mach cone flank. These points are fitted on both sides by a straight line to get the Mach angle. Errors of the angle are obtained by the fitting pro-cedure of the last step. We established method 4 as a less complex and more direct way to measure the Mach angle. The speed of sound can be calculated from the Mach relation Eq.(1). The inverse sine of the Mach angle θ is plotted against the speed of the supersonic particle. The velocity value of the regression line at 1/sin θ = 1 then gives the speed of sound:
1 sin θ
= 1 ⇒ vp= cs. (2)
The results of the measurements are displayed as plots of the Mach relation in the supp. material. The sound speed calculated from the angles obtained with the dif-ferent methods, is given in Tab. I. Still, none of the pro-posed methods uses all features of the Mach cone struc-ture (first compression zone, expansion zone and second compression zone). Using all the features would lead to a more stable algorithm and measurement. The manual measurement (Method 1) uses nearly all features of the Mach cone structure due to the human perception which itself is of disadvantage regarding the systematic error and reproducibility. The cross-correlation with fixed ro-tation center (Method 2) is the simplest computer-aided method. The cross-correlation with free center (Method 3) uses two features of the Mach cone structure but is the most complex method, and every step increases the uncertainties. Cone detection in perpendicular profiles (Method 4) uses a comprehensible algorithm, but needs the particle path as additional input. Also the detec-tion of the intensity peak from the first compression zone is challenging and can lead to large errors. Comparing the results and the difficulties of the methods, we recom-mend the cone detection in perpendicular profiles due to the easy and robust algorithm with credible uncertainty values.
Simulations for the plasma parameters were made to verify the results obtained above, and to estimate the particle charge. A 1D hybrid (fluid combined with particle-in-cell (PIC)) code41,45 and the 2D PIC simu-lation XOOPIC42 were used to get an estimation of the plasma parameters. The 1D simulation calculates the plasma parameters at different positions in the vertical direction between the electrodes and results in the plasma parameters displayed in Tab. II. The 2D PIC simulation results in values at the exact position of the supersonic
TABLE II. Plasma parameters (electron temperature Te, ion temperature Ti and electron density ne) simulated by using a 1D hybrid code41 resp. the 2D PIC simulation XOOPIC42
with a 30 VPP argon discharge at 60 mm electrode distance,
with a pressure of 2.85 Pa, and temperature of 0.0255 eV of the background gas.
parameters 1D 2D
Te(eV ) 2.7 - 2.9 2.9 - 3.1
Ti(10−2eV) 2.3 - 3.4 3.3 - 6.8
ne≈ ni(1014m−3) 1.2 - 2.2 0.7 - 1.1
FIG. 3. (Color online) Sound speed cs (blue curves on the left) and particle charge Z (red curves on the right) calculated from the plasma parameters Tab. II with Ref. 43 and 44. The experimental measurements result in a range of 18 − 22 mm/s for the speed of sound. This region is colored light blue in both plots.
particle. The ranges given in Tab. II are obtained by the averaged values of the plasma parameters at the cen-tral position compared to the respectively reproducible positions in the particular simulation. Though a fluid code might be not well suited for low gas pressures, the results are in a comparable range in both simulations. Both simulations do not include the interaction of the plasma with the microparticles, but the results can be used as an orientation. In order to calculate the speed of sound of the complex plasma cloud and the charge of the microparticles with the orbital motion limited (OML) theory43 we need to define appropriate parameters. We used the isotropic collisional cross-section from Ref. 44 to estimate the mean free path between charge-exchange collisions, resulting in λia= 0.72 mm for the neutral gas
pressure p = 2.85 Pa and Ti= Tn = 300 K. (Using newer
data from Ref. 46 results in λia = 0.67 mm for isotropic
scattering. We used λiafrom Ref. 44 since the values do
not differ significantly, although Ref. 44 is based on the older data of Ref. 47.) The electron density ne and the
ion temperature Ti are chosen as ranges from the
sim-ulated results in Tab. II. The microparticle density is fixed to nd = 82700 cm−3 and the electron temperature
to Te= 2.8 eV. The result is shown in Fig. 3.
The speed of sound from the simulated parameters is cs ≈ 2.0 ± 0.6 cm/s and the microparticle charge is
0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 v (m m s − 1)
FIG. 4. (Color online) Drift corrected average particle veloc-ity v in a section of 9.4 × 9.4 mm2 around the double Mach
cone structure. The location of this section is drawn in Fig. 5 c) as a yellow rectangle. We color-coded only a part of the velocity region for better contrast. Gray arrows have velocity values up to 18 mm/s. The black dot shows the position of the supersonic particle.
Z ≈ −5500 ± 1500 e. Theory and measurements match
considering the given uncertainty.
V. KINETIC ENERGY
We study the particle dynamics during the interaction with the supersonic particle. PTV39was used to get the position and the velocity of each microparticle. Vari-ations in density and velocity divergence directly show the Mach cone, see Ref. 37. The curl of the velocity field does not show a particular pattern. Since the complex plasma system was more in a fluid state than in a crys-talline state, we expect the waves to be dispersionless. Therefore, it matches with the theory25 to see compres-sional waves but no shear waves, as in Ref. 23. The ve-locity field is plotted in Fig. 4 as mean of 8 consecutive images. At the beginning, the supersonic particle and its first Mach cone pushes particles together in the first compression zone (Fig. 5 c)). The velocity field shows a v-shaped pattern with increased velocity perpendicular to the Mach cone. The direction of the velocity rotates in the expansion zone where particles start to move into the wake/cavity behind the supersonic particle. The sec-ond Mach cone then originates from the closure of the cavity as a second compression zone. In order to study the effect of the projectile on the cloud, we calculated kinetic energy of the microparticles. We improved the statistics by evaluating all 39 images of the sequence to-gether. This was done by transforming the particle co-ordinates into a local coordinate system with the probe
1 5 9 13 17 Ekin(eV) 1 3 5 7 9 11 Tkin(eV) a) b) q q c) d) e)
FIG. 5. (Color online) Effect of the supersonic particle on the energy a) and temperature distributions b) in the microparti-cle cloud. The positions of the micropartimicroparti-cles from 39 frames were adjusted to locate the supersonic particle in the center of the image, see white dot (with red border in color online). The total kinetic energy (Ekin Eq. 3 in a)) resp. kinetic
temperature (Tkin Eq. 4 in b)) is averaged by median over
the 20 nearest particles. The FoV of the resulting image is 16.5 × 21.2 mm2. Note that the heating occurs mainly behind
the supersonic particle and not in the wave fronts of the first Mach cone. Bottom figures show sections of 16.5 × 14.8 mm2
from the blurred overlay c) (like Fig. 2 a)), Ekin d) and Tkin
e). The double Mach cone structure is indicated by red lines and the position of the probe particle by a red dot. The yellow rectangle in c) shows the section of Fig. 4.
particle at the center. Then, particle tracks in a win-dow of ≈ 16.5 × 21.2 mm2around the probe particle were used. Fig. 5 shows maps of the total kinetic energy and of the so-called kinetic temperature, defined by:
total kinetic energy: Ekin= 1 2mv
(3) kinetic temperature: Tkin=
(v − hvi)2E (4) where hi indicates the median, m = 6.8 × 10−11g is the microparticle mass, and v the microparticle velocity. We used the median instead of the mean to reduce the influ-ence of outliers. For each point in Fig. 5, the kinetic en-ergy of the 20 closest particles was averaged. The Mach cone is clearly visible in the total kinetic energy while there is no cone in the kinetic temperature. Still, there is a heating behind the probe particle from the relax-ation of the wave. A small drift of the microparticles to the left was present during the experiment. This is the
cause of the asymmetric heating due to more collisions in the left expansion zone compared to the right. By com-paring the region of increased Ekin with the contours of increased density (see Fig. 5 c)-e)), we see that the to-tal kinetic Energy Ekin increases in front of the first and second Mach cone and stays increased behind it. The energy is low between the two Mach cones. The parti-cles are accelerated by the compression wave, and they slow down in the expansion zone. Hence, the changes in the total kinetic energy are driven by the overall drifts in the velocity field. In contrast, the kinetic temperature
Tkin increases mainly in the expansion zone in front of the second Mach cone. There the particle drift is small and collisions between decelerated particles behind the first Mach cone and accelerated particles in front of the second Mach cone distribute energy. The increase in ran-dom motions after these collisions is reflected by the in-crease of Tkin. We analyzed velocity histograms of the region outside and inside of the Mach cone by fitting 1D-Maxwell distributions to horizontal (x) and vertical (y) velocity distributions. The fits result in an independent measurement of the kinetic temperature, compare Fig. 6 and Fig. 5 b). The regions were chosen to be segments of an annulus with a predefined thickness of 2.35 mm (resp. 100 px). In Fig. 6, red data points are outside of the Mach cone, blue data points are inside of the Mach cone. The maximum horizontal distance to the probe particle was restricted to ≈ 8 mm. There is a higher kinetic tem-perature behind the probe particle than in front of it, and Tkindecreases with increasing distance to the super-sonic particle. The peak in the kinetic temperature in the y-direction at about 11 mm ahead of the probe particle is caused by hot particles at the border of the cloud at the beginning of the analyzed time sequence. In addition, the annulus region covers the outside of the microparticle cloud in the end of the sequence, resulting in less steady particles. Nevertheless, the heating of the microparticles behind the Mach cone is clearly visible.
A supersonic projectile particle caused a Mach cone in a 3D complex plasma cloud in microgravity conditions. Based on the continuous observation of the probe particle, the acoustic velocity across the cloud and perturbations are studied. Different methods for the measurement of Mach cones were proposed and used to analyze the data, leading to cs = 18.3 − 21.1 mm/s.
Calculation from OML theory43 based on simulated plasma parameters proposes a particle charge in the range of 4000 − 7000 electron charges. This results in a range from 16 − 26 mm/s for the speed of sound, which confirms the measurements. The high resolution of the data allows tracking of the microparticles by PTV to get the single particle trajectories around the supersonic particle. In Ref. 37, we calculated densities, divergence and curl of the velocities. Here we studied
FIG. 6. (Color online) Change in kinetic temperature Tkin
outside and inside of the Mach cone in x and y direction (dir.). The histogram of the x and y velocities of particles inside an annulus are fitted by the Maxwell distribution. Tkin(eV ) is
plotted against the distance to the supersonic particle. a) shows a sketch of the used areas at a distance of 3 mm. Red represents particles outside the Mach cone, blue particles in-side the Mach cone. b) Left: x (horizontal line) resp. y (dot) velocities in front of the projectile. Right: behind it. There is an increase in Tkinbehind the supersonic particle (located
at x=0), and a subsequent decrease of Tkin further away.
the energies of the microparticles. The analysis of the particle energies and temperatures yields new insights into the dynamics and energy distribution within the cloud in the disturbed vicinity of the projectile: Heating of the cloud takes place behind the Mach cone where the system tends to recover an equilibrium, but not at the position of the flanks. The heating is caused by random collisions of the microparticles which were accelerated by the supersonic projectile, either by its wave (Mach cone) or by direct interaction (displacement).
Further work could improve the algorithms such as bet-ter detection of the maximum by methods from signal processing. Furthermore, the single particle interaction of complex plasmas with waves (resp. Mach cones) cre-ates interesting new questions. For instance, what makes the differences between channeling, subsonic, and super-sonic probe particles since the structure of the heating seems similar to the heating of channeling particles48 although the underlaying processes differ considerably. Further investigations might include studying the heat-ing process in detail.
See Machconeorg.mpg for a video with a FoV of 16.5 × 21.2 mm2 following the supersonic particle. The Mach angle was measured on blurred and composed im-ages as described in the end of section II. A video of the event in the representation of these images is given in Machconeblur.mpg. The Mach relations as results of the 4 different methods described in section III is shown in Machrelations.eps.
We would like to thank Ingo Laut for carefully check-ing the manuscript. We would like to thank the whole Plasmalab team, espacially the engineers and mechan-ics, for continuously improving the setup. The Eko-plasma project is funded by the national aeronautics and space research center of the Federal Republic of Germany, Deutsches Zentrum für Luft- und Raumfahrt e.V., with funds from the German Federal Ministry for Economic Affairs and Energy, Bundesministerium für Wirtschaft und Energie, according to a resolution of the Deutscher Bundestag under grant FKZ 50WM1441.
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